Vorticity equation in index notation (curl of Navier-Stokes equation)

The trick is the following:

$$ \epsilon_{ijk} \frac{\partial u_m}{\partial x_j} \frac{\partial u_m}{\partial x_k} = 0 $$

by antisymmetry.

So you can rewrite

$$ \epsilon_{ijk} \frac{\partial u_m}{\partial x_j} \frac{\partial u_k}{\partial x_m} = \epsilon_{ijk} \frac{\partial u_m}{\partial x_j}\left( \frac{\partial u_k}{\partial x_m} - \frac{\partial u_m}{\partial x_k} \right) $$

Note that the term in the parentheses is something like $\pm\epsilon_{kml} \omega_l$

Lastly use the product property for Levi-Civita symbols

$$ \epsilon_{ijk}\epsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl} $$